A problem is slicewise polynomial (\(\mathrm {XP}\) ) for some parameter \(k\)  if it can be solved in time \(|I|^{f(k)}\)  for a computable function \(f\) . That is, the problem is solvable in polynomial time when \(k\)  is constant. A problem is called fixed-parameter tractable (\(\mathrm {FPT}\) ) for a parameter \(k\)  if it can be solved in time \(f(k) \cdot |I|^{\mathcal {O}(1)}\)  for a computable function \(f\) .
In general, \(\mathrm {FPT}\)  running times are preferable, because the degree of the polynomial is independent from the value of \(k\) , whereas in the case of \(\mathrm {XP}\)  running times, the degree of the polynomial might depend on the value of \(k\) .
If a problem is \(\mathrm {W}[1]\) -hard, then it is assumed to be fixed-parameter intractable.
The Clique problem when parameterized by size of the sought clique is an example for a \(\mathrm {W}[1]\) -hard problem.
For a detailed introduction into parameterized complexity we refer to the standard monographs [1]}, [2]}.
